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A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

%I A340854 #10 Apr 09 2021 09:41:21
%S A340854 1,2,4,6,8,10,14,16,20,22,26,28,32,34,38,44,46,52,58,62,64,68,74,76,
%T A340854 82,86,88,92,94,104,106,116,118,122,124,128,134,136,142,146,148,152,
%U A340854 158,164,166,172,178,184,188,194,202,206,212,214,218,226,232,236,244
%N A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.
%C A340854 Consists of 1 and all numbers that are even and have no odd divisor 1 < d <= n/d.
%e A340854 The sequence of terms together with their prime indices begins:
%e A340854       1: {}              44: {1,1,5}          106: {1,16}
%e A340854       2: {1}             46: {1,9}            116: {1,1,10}
%e A340854       4: {1,1}           52: {1,1,6}          118: {1,17}
%e A340854       6: {1,2}           58: {1,10}           122: {1,18}
%e A340854       8: {1,1,1}         62: {1,11}           124: {1,1,11}
%e A340854      10: {1,3}           64: {1,1,1,1,1,1}    128: {1,1,1,1,1,1,1}
%e A340854      14: {1,4}           68: {1,1,7}          134: {1,19}
%e A340854      16: {1,1,1,1}       74: {1,12}           136: {1,1,1,7}
%e A340854      20: {1,1,3}         76: {1,1,8}          142: {1,20}
%e A340854      22: {1,5}           82: {1,13}           146: {1,21}
%e A340854      26: {1,6}           86: {1,14}           148: {1,1,12}
%e A340854      28: {1,1,4}         88: {1,1,1,5}        152: {1,1,1,8}
%e A340854      32: {1,1,1,1,1}     92: {1,1,9}          158: {1,22}
%e A340854      34: {1,7}           94: {1,15}           164: {1,1,13}
%e A340854      38: {1,8}          104: {1,1,1,6}        166: {1,23}
%e A340854 For example, the factorizations of 88 are (2*2*2*11), (2*2*22), (2*4*11), (2*44), (4*22), (8*11), (88), none of which has odd minimum, so 88 is in the sequence.
%t A340854 Select[Range[100],Function[n,n==1||EvenQ[n]&&Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]=={}]]
%Y A340854 The version looking at greatest factor is A000079.
%Y A340854 The version for twice-balanced is A340656, with complement A340657.
%Y A340854 These factorization are counted by A340832.
%Y A340854 The complement is A340855.
%Y A340854 A033676 selects the maximum inferior divisor.
%Y A340854 A038548 counts inferior divisors.
%Y A340854 A055396 selects the least prime index.
%Y A340854 - Factorizations -
%Y A340854 A001055 counts factorizations.
%Y A340854 A045778 counts strict factorizations.
%Y A340854 A316439 counts factorizations by product and length.
%Y A340854 A339890 counts factorizations of odd length.
%Y A340854 A340653 counts balanced factorizations.
%Y A340854 - Odd -
%Y A340854 A000009 counts partitions into odd parts.
%Y A340854 A024429 counts set partitions of odd length.
%Y A340854 A026424 lists numbers with odd Omega.
%Y A340854 A066208 lists Heinz numbers of partitions into odd parts.
%Y A340854 A067659 counts strict partitions of odd length (A030059).
%Y A340854 A174726 counts ordered factorizations of odd length.
%Y A340854 Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.
%K A340854 nonn
%O A340854 1,2
%A A340854 _Gus Wiseman_, Feb 04 2021